For autonomous control systems, the control objectives need to be achieved robustly against system uncertainties.
Central to many control synthesis techniques for uncertain systems is backward reachablility analysis.
Given an uncertain control system and a set \(X_0\)  of target states, the backward reachable set consists of the states that can be steered into \(X_0\)  in finite time, regardless of the system uncertainties.
Being able to compute such sets is important to design controllers with safety or reachability objectives [1]}, [2]}, and is one building block for achieving more complicated control tasks [3]}, [4]}.
Whenever the exact computation is hard, an under-approximation can be still used to define a conservative control law.
A variety of approaches exists in the literature, including polyhedral computation [5]}, interval analysis [6]}, HJB method [7]} and polynomial optimization [8]}, just to name a few.
For linear dynamics with linear constraints, polyhedra can be used to represent the backward reachable sets as they are closed under linear transformation, Minkowski addition and subtraction, and can be computed leveraging linear optimization tools. However, its applicability is limited to low dimensional systems (typically, state dimension \(\le 4\) ) due to an expensive quantifier elimination step.
